A008957 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).

1, 1, 1, 1, 5, 1, 1, 14, 21, 1, 1, 30, 147, 85, 1, 1, 55, 627, 1408, 341, 1, 1, 91, 2002, 11440, 13013, 1365, 1, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1, 1, 285, 25194, 846260, 10787231

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• tabl

Comments

D. E. Knuth [1992] on page 10 gives the formula Sum n^(2m-1) = Sum_{k=1..m} (2k-1)! T(2m,2k) binomial(n+k, 2k) where T(m, k) is the central factorial number of the second kind. - Michael Somos, May 08 2018

Examples

			The triangle starts:
  1;
  1,  1;
  1,  5,    1;
  1, 14,   21,     1;
  1, 30,  147,    85,     1;
  1, 55,  627,  1408,   341,    1;
  1, 91, 2002, 11440, 13013, 1365, 1;
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a).
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Links

• Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
• F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. [From _N. J. A. Sloane_, Jan 02 2013]
• D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
• D. E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222, Jul 1992. See bottom of page 10. [From _Michael Somos_, May 08 2018]
• Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.

Crossrefs

Row reversed version of A036969. (0,0)-based version: A269945.

Cf. A008955.

Programs

Haskell

a008957 n k = a008957_tabl !! (n-1) (k-1)
a008957_row n = a008957_tabl !! (n-1)
a008957_tabl = map reverse a036969_tabl
-- Reinhard Zumkeller, Feb 18 2013

Maple

A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),j=1..k); end; # Gives rows of triangle in reversed order

Mathematica

t[n_, n_] = t[n_, 1] = 1;
t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k];
Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* Jean-François Alcover, Jun 16 2011 *)

PARI

{T(n, k) = if( n<1 || k>n, 0, n==k || k==1, 1, T(n-1, k-1) + k^2 * T(n-1, k))}; \\ Michael Somos, May 08 2018

Sage

def A008957(n, k):
    m = n - k
    return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m))
for n in (1..7): print([A008957(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018

Formula

From Michael Somos, May 08 2018: (Start)

T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1.

E.g.f.: x^2 * (cosh(sinh(y*x/2) / (x/2)) - 1) = (1*x^2)*y^2/2! + (1*x^2 + 1*x^4)*y^4/4! + (1*x^2 + 5*x^4 + 1*x^6)*y^6/6! + (1*x^2 + 14*x^4 + 21*x^6 + 1*x^8)*y^8/8! + ... (End)

Extensions

More terms from Vladeta Jovovic, Apr 16 2000